Hamilton's Math To Build On - copyright 1993

45° Right Triangles

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* Marking 45° Right Triangles

Every 45° right triangle has two equal leg lengths and a hypotenuse length of 1.4142 times the length of one of those legs. To verify this, follow the instructions below to first draw a 45° right triangle.

First: From a horizontal line, draw a perpendicular line then bisect it to make a 45° angle.

Second: Draw two more perpendicular lines from the horizontal line so that each perpendicular line crosses the 45° line. Notice that each perpendicular creates a different 45° right triangle.

Third: Measure the length of each leg of each 45° right triangle.

For each triangle, there should be two equal legs and a hypotenuse with a length of 1.4142 times the length of a leg.

Right triangles can be inscribed in a circle using the diameter of the circle as the hypotenuse for each triangle.

First: Draw any size circle and a diameter in that circle.

Second: Draw a straight line from the point where the circle and the diameter meet to any other point on the circle.

Third: Draw another straight line from the second point on the circle to the other endpoint of the diameter.

You have drawn a right triangle. Notice that the diameter is the hypotenuse of the right triangle.

Fourth: Draw four more right triangles in the same circle.

Measure the sides of each right triangle and notice that the hypotenuse is always the longest side.

Skilled carpenters use this knowledge to draw circles with their framing squares. They set two nails with a distance between them equal to the diameter of a circle they need. They then place a framing square between the two nails and mark the heel of the square as they turn the square between the nails.

Notice that the diameter of the circle and the square create a right triangle at each position. Also notice that this procedure is not exactly accurate since the square never touches the center point of the nail. This is an example of accuracy needed on a job. Even though this technique is accurate enough for many rough carpentry jobs, there are many times in carpentry and other trades that this degree of accuracy is not acceptable.

On to Drawing Right Triangles Using Calculated Lengths

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18 September 1995
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